The Prophet Joseph Smith explained about the Holy Ghost coming down and abiding on Jesus: “The sign of the dove was instituted before the creation of the world, a witness for the Holy Ghost, and the devil cannot come in the sign of a dove. The Holy Ghost is a personage, and is in the form of a personage. It does not confine itself to the form of the dove, but in sign of the dove. The Holy Ghost cannot be transformed into a dove; but the sign of a dove was given to John to signify the truth of the deed, as the dove is an emblem or token of truth and innocence.” 30
See also commentary at 2 Nephi 31:8.